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Grötzsch theorems

Various results on conformal and quasi-conformal mappings obtained by H. Grötzsch . He developed the strip method, which is the first general form of the method of conformal moduli (cf. Extremal metric, method of the; Strip method (analytic functions)), and used it in his systematic study of a large number of extremal problems of conformal mapping of multiply-connected (including infinitely-connected) domains, including the problems of the existence, uniqueness and geometric properties of extremal mappings. A few of the simpler Grötzsch theorems are presented below.

Of all univalent conformal mappings of a given annulus under which the unit circle is mapped onto itself, the maximum diameter of the image of the circle is attained if and only if the boundary component is a rectilinear segment with its centre at the point . A similar result is valid for multiply-connected domains.

Out of all univalent conformal mappings of a given multiply-connected domain with expansion at infinity and normalization at a given point , the maximum of , and the maximum (minimum) of at a given point , , are attained only on mappings that map each boundary component of , respectively, to an arc of a circle with centre at the point , or to an arc of an ellipse (hyperbola) with foci at the points and . In each one of these problems the extremal mapping exists and is unique. In this class of mappings, for a given , the disc

is the range of the function . Each boundary point of this disc is a value of on a unique mapping in the class under study with specific geometric properties.

Grötzsch was the first to propose a form of representation of a quasi-conformal mapping, and to apply to such a mappings many extremal results which he had formerly obtained for conformal mappings.